Abstract
We show that a realization of the Laplace operator Au := u′′ with general nonlocal Robin boundary conditions α j u′(j) + β j u(j) + γ 1–j u(1 − j) = 0, (j = 0, 1) generates a cosine family on L p(0, 1) for every \({p\,{\in}\,[1,\infty)}\). Here α j , β j and γ j are complex numbers satisfying α 0, α 1 ≠ 0. We also obtain an explicit representation of local solutions to the associated wave equation by using the classical d’Alembert’s formula.
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References
Arendt W. et al.: Vector-valued Laplace Transforms and Cauchy Problems. Birkhäuser, Basel (2001)
Bobrowski A.: Generation of cosine families via Lord Kelvin’s method of images. J. Evol. Equ. 10, 663–675 (2010)
R. Chill, V. Keyantuo, and M. Warma, Generation of cosine families on L p(0, 1) by elliptic operators with Robin boundary conditions, Functional analysis and evolution equations, 113–130, Birkhäuser, Basel, 2008.
Fattorini H.O.: Second Order Linear Differential Equations in Banach Spaces. North-Holland Publishing Co., Amsterdam (1985)
Goldstein J.A.: Semigroups of Linear Operators and Applications. Oxford University Press, New York (1985)
Kisyński J.: On cosine operator functions and one-parameter groups of operators. Studia Math. 44, 93–105 (1972)
Xiao T.-J., Liang J.: Second order differential operators with Feller-Wentzell type boundary conditions. J. Funct. Anal. 254, 1467–1486 (2008)
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Alvarez-Pardo, E., Warma, M. The one-dimensional wave equation with general boundary conditions. Arch. Math. 96, 177–186 (2011). https://doi.org/10.1007/s00013-010-0209-y
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DOI: https://doi.org/10.1007/s00013-010-0209-y